Convex Duality in Constrained Portfolio Optimization

TLDR

The study is set in a continuous‑time Ito process model of asset prices. The paper investigates maximizing expected utility from terminal wealth and/or consumption under portfolio constraints to a closed convex set. The authors employ continuous‑time martingale methods, convex analysis, and duality theory, embedding the constrained problem into a family of unconstrained ones to prove existence of optimal strategies. They establish equivalent optimality conditions, derive explicit feedback solutions for certain utilities and deterministic coefficients, and extend the results to incomplete markets, short‑selling constraints, and asymmetric borrowing/lending rates.

Abstract

We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of $\mathscr{R}^d$. The setting is that of a continuous-time, Ito process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory.